Sorry about the emails I sent that had nothing to do with the compressor
The wiring diagram I have is terrible but a simple test is to pull the connector off the switch. If the compressor engages, the swtich is bad.
You should also test the swtich on the drier with the blue/red wire. Jump the ends and see if that gets the compressor going. If so, the pressure switch is bad. I think you have to evacuate the system to replace the switch.
Other suspects are the cabin temp sensor, evaporator temp sensor, temp controller, pushbutton unit and the Klima relay. Diagnosis of these items is not so simple.
Have you tried with the temp wheel set to min - to the click? Have you tried in defrost mode? Are you sure you have a proper refrigerant charge?